Analysis II Prelim topics
Analytic functions: power series, exponential and logarithm functions, Moebius transformations, the Riemann sphere.
Cauchy’s theorem: Goursat’s proof, homotopic curves, winding number of a closed curve about a point, the Cauchy integral formulas, Liouville’s theorem, isolated singularities, Casorati-Weierstrass theorem, open mapping theorem, maximum principle, Morera’s theorem, and Schwarz reflection principle, residue theorem, the argument principle, Rouche’s theorem, the evaluation of definite integrals, Schwarz’s lemma, zeros of analytic functions.
Harmonic functions: conjugate functions, mean value property.
Meromorphic functions: Isolated singularities and their classification, Laurent series expansion, Casorati-Weierstrass theorem, Picard’s theorems (statements only), principal part of a meromorphic function at an isolated singularity, partial fraction expansion. Also, meromorphic functions viewed as analytic maps into the Riemann sphere.
Entire functions: infinite products, Jensen’s formula, Weierstrass product theorem, Hadamard factorization theorem.
Conformal mappings: Elementary mappings, mappings by Moebius transformations, the Riemann mapping theorem (statement only).
- Hilbert spaces: orthogonality, orthogonal decompositions, Riesz Representation theorem, Bessel's inequality, orthonormal bases, Parseval's identity
- Fourier transform: Tempered distributions, the Fourier transform on L2, Plancherel's theorem, Sobolev spaces
- Ahlfors, L. (1979). Complex Analysis, New York, McGraw Hill (3rd edition).
- Conway, J. B. (1978) Functions of One Complex Variable I (Graduate texts in Mathematics-vol. 11), Springer (2nd edition)
- Stein, E.M. and Sharkarchi, R. (2003). Complex Analysis, Princeton University Press.
- Titchmarsh, E. C. (1939) The Theory of Functions, Oxford University Press (2nd edition).
- S. Axler, “Measure, integration & real analysis”
- F.G. Friedlander and M. Joshi, “Introduction to the theory of distributions”